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I will start describing some basics of the graph Laplacian eigenvectors of a given graph and their properties. In particular, I will describe the peculiar phase transition/localization phenomena of such eigenvectors observed on a certain type of graphs (e.g., dendritic trees of neurons). Then, I will describe how to construct wavelet packets on a given graph including the Haar-Walsh basis dictionary using the graph Laplacian eigenvectors. As an application of such basis dictionaries, I will discuss efficient approximation of functions given on graphs.

The Turan number ex(n,H) of an r-graph H is the largest size of an n-vertex r-graph that does not contain H. The famous Erdos-Sos conjectrure concerns theTuran number of a tree T of k vertices. The difficulty lies in the fact thatthere could be very different extremal families, disjoint cliques of sizes k-1or in some cases a graph with (k-2)/2 vertices of degree n-1.